A size-balanced binary tree is a binary tree in which for every node the difference between the number of nodes in the left and right subtree is at most $1$. The distance of a node from the root is the length of the path from the root to the node. The height of a binary tree is the maximum distance of a leaf node from the root.

Prove, by using induction on h, that a size-balance binary tree of height $h$ contains at least $2^h$ nodes.

In a size-balanced binary tree of height $h \geqslant 1$, how many nodes are at distance $h-1$ from the root? Write only the answer without any explanations.